| import OAI.NumberTheory.Jacobsthal.Probability.ThirdInnerKernel |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| section |
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| open _root_.Set _root_.Erdos970.Set _root_.MeasureTheory _root_.Erdos970.MeasureTheory |
| namespace ErdosOmissionBindings |
| open ErdosContinuousOmission ErdosContinuousBoundary ErdosBoundaryLoss |
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| noncomputable def thirdLowerIntegral (p : ℝ×ℝ) : ℝ := |
| ∫ t : ℝ in 1..baseCutoff p.2,clampedThird p.1 p.2 t |
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| noncomputable def thirdUpperIntegral (p : ℝ×ℝ) : ℝ := |
| ∫ x : ℝ in baseCutoff p.1..2,clampedThird x p.1 p.2 |
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| theorem thirdLowerIntegral_continuous : Continuous thirdLowerIntegral := by |
| let F : (ℝ×ℝ) → ℝ → ℝ := fun p t => clampedThird p.1 p.2 t |
| have hF : Continuous F.uncurry := by |
| change Continuous ((fun q : ℝ×ℝ×ℝ => clampedThird q.1 q.2.1 q.2.2) ∘ |
| (fun p : (ℝ×ℝ)×ℝ => (p.1.1,p.1.2,p.2))) |
| exact clampedThird_continuous.comp |
| (continuous_fst.fst.prodMk (continuous_fst.snd.prodMk continuous_snd)) |
| exact continuous_variable_upper_integral F hF (fun p => baseCutoff p.2) |
| (baseCutoff_continuous.comp continuous_snd) |
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| theorem thirdUpperIntegral_continuous : Continuous thirdUpperIntegral := by |
| let F : (ℝ×ℝ) → ℝ → ℝ := fun p x => clampedThird x p.1 p.2 |
| have hF : Continuous F.uncurry := by |
| change Continuous ((fun q : ℝ×ℝ×ℝ => clampedThird q.1 q.2.1 q.2.2) ∘ |
| (fun p : (ℝ×ℝ)×ℝ => (p.2,p.1.1,p.1.2))) |
| exact clampedThird_continuous.comp |
| (continuous_snd.prodMk (continuous_fst.fst.prodMk continuous_fst.snd)) |
| have h2 := continuous_variable_upper_integral F hF (fun _ => (2:ℝ)) continuous_const |
| have hc := continuous_variable_upper_integral F hF (fun p => baseCutoff p.1) |
| (baseCutoff_continuous.comp continuous_fst) |
| have he (p : ℝ×ℝ) : (∫ x : ℝ in 1..2,F p x)-(∫ x : ℝ in 1..baseCutoff p.1,F p x)=thirdUpperIntegral p := by |
| have hf : Continuous (F p) := by |
| change Continuous (F.uncurry ∘ (fun x : ℝ => (p,x))) |
| exact hF.comp (continuous_const.prodMk continuous_id) |
| have hi := intervalIntegral.integral_add_adjacent_intervals |
| (hf.intervalIntegrable (μ := volume) 1 (baseCutoff p.1)) |
| (hf.intervalIntegrable (μ := volume) (baseCutoff p.1) 2) |
| change (∫ x in 1..2,F p x)-(∫ x in 1..baseCutoff p.1,F p x)=∫ x in baseCutoff p.1..2,F p x |
| linarith |
| exact (h2.sub hc).congr he |
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| theorem thirdLowerIntegral_bound (p : ℝ×ℝ) : |thirdLowerIntegral p| ≤ 64 := by |
| have h := intervalIntegral.norm_integral_le_of_norm_le_const (a:=1) (b:=baseCutoff p.2) |
| (C:=64) (f:=fun t => clampedThird p.1 p.2 t) |
| (fun t _ => by simpa only [Real.norm_eq_abs] using clampedThird_bound p.1 p.2 t) |
| have hc := baseCutoff_bounds p.2 |
| change |thirdLowerIntegral p| ≤ 64 |
| rw [Real.norm_eq_abs,abs_of_nonneg (by linarith : 0 ≤ baseCutoff p.2-1)] at h |
| exact h.trans (by nlinarith) |
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| theorem thirdUpperIntegral_bound (p : ℝ×ℝ) : |thirdUpperIntegral p| ≤ 64 := by |
| have h := intervalIntegral.norm_integral_le_of_norm_le_const (a:=baseCutoff p.1) (b:=2) |
| (C:=64) (f:=fun x => clampedThird x p.1 p.2) |
| (fun x _ => by simpa only [Real.norm_eq_abs] using clampedThird_bound x p.1 p.2) |
| have hc := baseCutoff_bounds p.1 |
| change |thirdUpperIntegral p| ≤ 64 |
| rw [Real.norm_eq_abs,abs_of_nonneg (by linarith : 0 ≤ 2-baseCutoff p.1)] at h |
| exact h.trans (by nlinarith) |
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| theorem thirdLowerIntegral_eq {x u : ℝ} (hx : x ∈ Icc (1:ℝ) 2) (hu : u ∈ Icc (1:ℝ) 2) : |
| thirdLowerIntegral (x,u)=∫ t : ℝ in 1..u,loss3Kernel t u x := by |
| rw [thirdLowerIntegral,baseCutoff_on hu.1 hu.2] |
| apply intervalIntegral.integral_congr |
| intro t ht |
| rw [uIcc_of_le hu.1] at ht |
| exact clampedThird_eq hx hu ⟨ht.1,ht.2.trans hu.2⟩ |
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| theorem thirdUpperIntegral_eq {u t : ℝ} (hu : u ∈ Icc (1:ℝ) 2) (ht : t ∈ Icc (1:ℝ) 2) : |
| thirdUpperIntegral (u,t)=∫ x : ℝ in u..2,loss3Kernel t u x := by |
| rw [thirdUpperIntegral,baseCutoff_on hu.1 hu.2] |
| apply intervalIntegral.integral_congr |
| intro x hx |
| rw [uIcc_of_le hu.2] at hx |
| exact clampedThird_eq ⟨hu.1.trans hx.1,hx.2⟩ hu ht |
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| end ErdosOmissionBindings |
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| end |
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| end Erdos970 |
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| end OAI |
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